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If x and y are positive integers , is x^y + (x^x)(y^y) odd?

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Bunuel
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If x and y are positive integers , is x^y + (x^x)(y^y) odd? [#permalink] New post   17 Nov 2021, 07:00
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If x and y are positive integers , is \(x^y + (x^x)(y^y)\) odd?

(1) x + 2y is odd
(2) y + 2x is odd
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If x and y are positive integers , is x^y + (x^x)(y^y) odd? [#permalink] New post   17 Nov 2021, 07:12
Bunuel wrote:
If x and y are positive integers , is \(x^y + (x^x)(y^y)\) odd?

(1) x + 2y is odd
(2) y + 2x is odd
Is \(x^y + (x^x)(y^y)\) odd?
For this to be odd, either of \(x^y\) & \((x^x)(y^y)\) needs to be odd and the other even.
But, since 'x' is present on both sides, 'x' can not be even, otherwise both the terms would be even. Thus, x has to be odd, and y has to be even. If we can prove this, it will answer our question: Is \(x^y + (x^x)(y^y)\) odd?

(1) x + 2y is odd.
For this to be odd, x has to be odd. We do not know the nature of y. y can be either even or odd. Insufficient.

(2) y + 2x is odd.
For this to be odd, y has to be odd. x can be even or odd. Insufficient.

(1) + (2),
x and y both are odd. Thus, \(x^y + (x^x)(y^y)\) will be even, because odd + odd = even.
Thus is sufficient to answer the question, Is \(x^y + (x^x)(y^y)\) odd? No. (C)
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Re: If x and y are positive integers , is x^y + (x^x)(y^y) odd? [#permalink] New post   17 Nov 2021, 07:37
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If x and y are positive integers , is x^y+(x^x)(y^y) odd?

Essentially asking two possibilities
1. If x is odd and y is even then x^y+(x^x)(y^y) becomes odd + odd x even = odd
or
2. If y is odd. In this case we have two cases
a. If x is even then x^y+(x^x)(y^y) becomes even + odd x even = even
b. If x is odd then x^y+(x^x)(y^y) becomes odd + odd x odd = even
Hence we can definitely prove that x^y+(x^x)(y^y) is not odd

irrespective of x being odd or even the answer is always even
(1) x + 2y is odd
Here x is odd but we don't know anything about y. Insufficient

(2) y + 2x is odd
Here y is odd.
As noted above, when y is odd, whether x is odd or even, the expression x^y+(x^x)(y^y) is always even. Hence this is sufficient to prove a Yes/No question

Option B
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Re: If x and y are positive integers , is x^y + (x^x)(y^y) odd? [#permalink] New post   17 Nov 2021, 07:38
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PyjamaScientist wrote:
Thus, x has to be odd, and y has to be even. If we can prove this, it will answer our question: Is \(x^y + (x^x)(y^y)\) odd?

(2) y + 2x is odd.
For this to be odd, y has to be odd. x can be even or odd. Insufficient.


You've determined at the outset the conditions under which the answer to the question will be 'yes'. But if we learn from one statement that those conditions are not satisfied, we can then be sure the answer to the question is 'no', and that's all we need in DS: to be certain of the answer to the question. That's what happens with Statement 2: we learn y is odd, so y is certainly not even, and so the answer to the question must be 'no', and Statement 2 is sufficient.

The exponents in this question aren't going to change anything about evenness or oddness, so the question is really asking if x + xy is odd, or if x(y + 1) is odd. Statement 1 tells us x is odd, but we don't know about y+1. Statement 2 tells us y is odd, so y+1 is even, and x(y+1) must be even, so the answer to the question must be 'no', and the answer is B.
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Re: If x and y are positive integers , is x^y + (x^x)(y^y) odd? [#permalink] New post   02 Jan 2024, 00:10
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Re: If x and y are positive integers , is x^y + (x^x)(y^y) odd? [#permalink]
02 Jan 2024, 00:10
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